This section reviews the framework developed in Benaych-Georges and Rao Nadakuditi's work [1]}, [2]} addressing the \(\beta _n \rightarrow \beta > 0 \)  setting, followed by a discussion of adaptations required to study the \(\beta _n \rightarrow 0\)  setting. We assume \(\varepsilon _{i,n} = 0\) , meaning \(\theta _i = \tau _i \beta _n^{1/4}\) .
The Stieltjes transforms of the ESD of \(S_n\)  and the Marchenko-Pasteur law with parameter \(\beta _n\) , introduced in Section REF , are defined as follows:
\(\phantom{\,,} s_n(z) &= \int \frac{1}{\lambda - z} dF_n(\lambda ) = \frac{1}{n} \mathrm {tr}(S_n - zI_n)^{-1} \,, \nonumber \\ \phantom{\,.}\bar{s}_n(z) &= \int \frac{1}{\lambda - z} d \hspace{0.83328pt}\overline{\hspace{-0.83328pt}F\hspace{-0.83328pt}}\hspace{0.83328pt}_n(\lambda ) = \frac{1 - \beta _n - z + \sqrt{(1 - \beta _n - z)^2 - 4 \beta _n }}{2\beta _n z} \,, \) 
